With the exploration measure and the support of the kernel, we select a class of basis functions \(\lbrace \phi _i\rbrace _{i=1}^n\)  and a range of \(n\)  for the hypothesis space \(\mathcal {H}_n =\mathrm {span}\lbrace \phi _i\rbrace _{i=1}^n\) . The basis function can be either global basis functions such as Bernstein polynomials as those used in [1]}, [2]} and trigonometric functions, or local basis functions such B-spline polynomials (see Appendix REF  for a brief introduction). We focus on local basis functions because they are more flexible to adaptive to local structure of the kernel. To set the range for \(n\) , we note that the mesh points of the kernel's independent variable explored by data are \(\lbrace k\Delta x: k=1,\ldots , \lfloor {\frac{R}{\Delta x}}\rfloor \rbrace \) . Meanwhile, the basis function should be linearly independent in \(L^2(\rho _N^J)\)  so that the basis matrix
\(B_n = ( \langle {\phi _i,\phi _j}\rangle _{L^2(\rho _N^J)})_{1\le i,j\le n} \in \mathbb {R}^{n\times n}\) 
